Timeline for Partitioning a convex $n$-polygon
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 2, 2021 at 11:15 | history | edited | TheVal | CC BY-SA 4.0 |
deleted 7 characters in body
|
| May 2, 2021 at 10:14 | vote | accept | TheVal | ||
| S Apr 30, 2021 at 20:53 | history | suggested | Jukka Kohonen | CC BY-SA 4.0 |
attempted to clarify the question, as I understand from the comments. In particular, parts are NOT disjoint but their interiors are.
|
| Apr 30, 2021 at 13:40 | review | Suggested edits | |||
| S Apr 30, 2021 at 20:53 | |||||
| Apr 30, 2021 at 13:20 | answer | added | Joseph O'Rourke | timeline score: 3 | |
| Apr 30, 2021 at 12:35 | review | Close votes | |||
| May 12, 2021 at 3:02 | |||||
| Apr 30, 2021 at 12:16 | comment | added | Ilya Bogdanov | In this case, for $K=5$ you will get an embedding of $K_5$ in the plane. | |
| Apr 30, 2021 at 10:59 | comment | added | TheVal | @JosephO'Rourke Yep, yours is a solution, as the partition share the central vertex $x$. I should add another question with a stricter requirement, that all $p_i$ must share at least one edge with all other parts. | |
| Apr 30, 2021 at 10:48 | history | edited | TheVal | CC BY-SA 4.0 |
added 58 characters in body
|
| Apr 30, 2021 at 10:47 | comment | added | TheVal | @JukkaKohonen Absolutely yes, and I'm trying to visualize Joseph solution, I'll tell in the comments asap. If that works, it surely will be a neat answer! | |
| Apr 30, 2021 at 10:38 | comment | added | Jukka Kohonen | Thanks, it is clearer now. Is it also so that the INTERIORS of the parts must be pairwise disjoint? (Otherwise surely they cannot share a vertex or an edge.) Still it seems Joseph's comment would provide an easy solution, so is this what you are after, or something else? | |
| Apr 30, 2021 at 10:35 | comment | added | TheVal | @JukkaKohonen I've edited my question. I apologize for the bad writing, and I hope I clarified a bit. I could add some diagrams if it helps to drive the point of my question home | |
| Apr 30, 2021 at 10:33 | history | edited | TheVal | CC BY-SA 4.0 |
added 160 characters in body
|
| Apr 30, 2021 at 9:52 | comment | added | TheVal | @JukkaKohonen Terribly sorry. I'll clarify right away: YES $p_i$s are pairwise disjoint, and YES $p_i$s are poligonal parts (subsets) of $P$, and $\mathcal{Z}_K(P)$ is a partition into $K$ parts and NOT the set of all such partitions | |
| Apr 30, 2021 at 6:18 | comment | added | Jukka Kohonen | Fedor, my guess is the OP means that their interiors must have empty intersection. This is another thing that should be clarified in the question. | |
| Apr 30, 2021 at 5:56 | comment | added | Fedor Petrov | Wait, for $K=2$ the intersection of parts must be empty but they should have a common edge or vertex. How is this possible? | |
| Apr 30, 2021 at 5:40 | history | edited | Francesco Polizzi |
edited tags
|
|
| Apr 30, 2021 at 5:26 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
deleted 47 characters in body
|
| Apr 30, 2021 at 3:12 | comment | added | Wlod AA | Let $P=S^n$ be the boundary of an $(n+1)$-simplex. The intersection of the $n+2$ main faces is empty, the union is the entire $S^n$, and every two intersect at their common $(n-1)$-dimensional face. | |
| Apr 30, 2021 at 3:05 | history | edited | Wlod AA | CC BY-SA 4.0 |
partitions are something else
|
| Apr 30, 2021 at 0:37 | comment | added | Joseph O'Rourke | Why is this not a solution? Let $x$ be any strictly interior point of $P$. Draw spokes from $x$ to the boundary of $P$, with adjacent spokes separated by $2 \pi /K$. The resulting pieces $p_i$ each share $x$. | |
| Apr 30, 2021 at 0:21 | comment | added | Jukka Kohonen | Could you clarify your notation slightly? I believe you want to specify that the $p_i$ are pairwise disjoint, instead of saying the intersection of all of them is empty. Also I get confused by what is a "part", what is "partition" and what is "partitioning". Are $p_i$ polygonal parts (subsets) of $P$? Is ${\cal Z}_K(P)$ a partition into $K$ such polygonal parts, as your notation says, or is it the set of all such partitions? | |
| Apr 29, 2021 at 23:53 | history | asked | TheVal | CC BY-SA 4.0 |